HAZARDOUS BACKWARD IN TIME CON- TINUATION IN NONLINEAR PARABOLIC - - PowerPoint PPT Presentation

HAZARDOUS BACKWARD IN TIME CON- TINUATION IN NONLINEAR PARABOLIC EQUATIONS AND DEBLURRING NON- LINEARLY BLURRED IMAGERY. by ALFRED S. CARASSO, ACMD

Identify sources of groundwater pollution Solve Advection Dispersion Equation back- ward in time, given present state g(x, y): Ct = ∇.{D∇C} − ∇.{vC}, 0 < t ≤ T, C(x, y, T) = g(x, y). (1)

DEBLURRING HUBBLE GALAXY IMAGES

Original Tadpole Deblurred Image

Solve logarithmic diffusion equation back- ward in time, given blurred image g(x, y): wt = −

• λ log{1 + γ(−∆)β}
• w,

0 < t ≤ T, w(x, y, T) = g(x, y). (2)

BACKWARD PARABOLIC EQNS VERY ILL-POSED BACKWARD UNIQUENESS AND STABILITY ??? Linear or nonlinear parabolic equn wt = Lw, t > 0. Let w1(x, t), w2(x, t) be any two solutions. Let F(t) = w1(., t) − w2(., t) 2 (L2 norm) Logarithmic Convexity techniques lead to F(t) ≤ {F(0)}1−µ(t){F(T)}µ(t), 0 ≤ t ≤ T. H¨

• lder exponent µ(t) satisfies 0 ≤ µ(t) ≤ 1, µ(T) = 1,

µ(0) = 0, µ(t) > 0 for t > 0, and µ(t) ↓ 0 as t ↓ 0. If solutions satisfy prescribed bound w(., 0) 2≤ M, then F(0) ≤ 2M. Hence F(T) = 0 ⇒ F(t) = 0 on [0, T]. This implies Backward Uniqueness. Fix any small δ > 0. Then F(T) ≤ δ ⇒ F(t) ≤ 2M 1−µ(t)δµ(t)

• n [0, T]. This implies Backward Stability.

Precarious backward stability F(t) ≤ 2M1−µ(t)δµ(t).

Autonomous, selfadjoint ⇒ µ(t) = t/T. Otherwise, µ(t) sublinear in t, possibly with fast decay to zero.

Autonomous, linear self adjoint problem Nonlinear problem

Behavior of Holder exponent in backward problems

wt = ectwxx, 0 < x < π, wx(0, t) = wx(π, t) = 0, t ≥ 0. µ(t) = {1 − exp(ct)}/{1 − exp(cT)}, 0 ≤ t ≤ T.

wt = 0.05 e(0.025x+0.05t)wx

• x + {sin(4πx)} wx, t ≥ 0,

wred(x, 0) = e3x sin2(3πx), w(−1, t) = w(1, t) = 0.

t=0 t=0 t=1 Effective non uniqueness in non self adjoint case

Either red or green initial values at t=0, terminate on black curve at t=1 to within 1.4E−3 pointwise, and L2 relative error = 2.3E−4.

THE FUNNEL CONJECTURE

FUNNEL EFFECT IN PARABOLIC EQUATIONS

EXPLORE 2D BACKWARD CONTINUATION

Nonlinear parabolic initial value problem on Ω × (0, T) wt = γr(w)∇.{q(x, y, t)∇w} + awwx, +b(wcos2w)wy, w(x, y, 0) = g(x, y). Ω square 0 < x, y < 1, T > 0, Zero Neumann on ∂Ω. r(w) = exp(0.025w), q(x, y, t) = e10t(1 + 5e2ysinπx). γ = 8.5×10−4, a, b constants ≥ 0, yet to be prescribed. FIND INTERESTING INITIAL VALUES g(x, y) ????

EXAMPLES OF DATA

Gaussian inital data More interesting data

MORE EXAMPLES OF DATA 8 bit images with values between 0 and 255.

MRI Brain image Pyramids image

NONLINEAR PARABOLIC IMAGE BLURRING wt = γr(w)∇.{q(x, y, t)∇w} + awwx + b(wcos2w)wy Forward problem well-posed in L2(Ω). For any initial value w(x, y, 0) = g(x, y), unique solution w(x, y, T) ex- ists at time T. Nonlinear solution operator ΛT well- defined on L2(Ω) by formula ΛTg(x, y) = w(x, y, T). w(x, y, T) ∈ very restricted class of smooth functions. Restrict attention to 256×256 images. Using centered space differencing, with ∆x = ∆y = 1/256, ∆t = 3.0E−7, march forward 400 time steps to T = 1.2E−4, using following explicit finite difference scheme W n+1 = W n + ∆tγR(W n)∇.{Qn∇W n} + aW nW n

x

+ b(W ncos2W n)W n

y ,

n = 0, 399, W 0 = g(x, y). (3) Discrete nonlinear solution operator ΛT

d well-defined

• n 256 × 256 images by formula ΛT

d W 0 = W 400.

NONLINEAR BACKWARD CONTINUATION Very little known about nonlinear multidimensional backward in time parabolic equations. Computation of such problems lies in uncharted waters. wt = γr(w)∇.{q(x, y, t)∇w} + awwx + b(wcos2w)wy Solution operator ΛTg(x, y) = w(x, y, T), defined on L2. VanCittert iterative procedure Originated in Spectroscopy in 1930’s. 1D Linear con- volution integral equations, with explicitly known ker- nels, assumed to have positive Fourier transforms. With f(x, y) ≈ w(x, y, T), fixed 0 < λ < 1, consider hm+1(x, y) = hm(x, y)+λ

• f(x, y) − ΛThm(x, y)
• , m ≥ 1,

where h1(x, y) = λf(x, y). Convergence ⇒ ΛTh∞(x, y) = f(x, y). Mathematically impossible with noisy data f(x, y).

Nevertheless, Van Cittert is valuable tool in large va- riety of 2D nonlinear backward equations. In many cases, L∞ norm of residual, f − ΛThm ∞, decays quasi-monotonically to small value after a finite num- ber N of iterations, and hN(x, y) is useful approximation to w(x, y, 0). Cases also exist where method fails !!!. VANCITTERT NONLINEAR DEBLURRING Given 256 × 256 blurred image f(x, y) ≈ w(x, y, T), use above centered difference scheme to do ∗ ∗ ∗ Hm+1 = Hm + λ

• f(x, y) − ΛT

d Hm

, m ≥ 1, ∗ ∗∗ where H1 = λf(x, y). Here, ΛT

d Hm means marching ex-

plicit difference scheme 400 time steps forward, using image Hm(x, y) as initial data W 0. Self-regularizing property of VanCittert Deblurring Iterative process recovers low frequency information in first several iterations. Many more iterations needed to recover high-frequency information. Interactively stopping iteration after finitely many steps, can recover deblurred image relatively free from noise. (Unless method fails for that image !!!)

Diagnostic image metrics L1 and TV norms Blurring and deblurring experiments will use following norms to evaluate results f 1=

• (256)−2 256

x,y=1 |f(x, y)|

• ,

∇f 1= (256)−2 255

x,y=1

• {fx(x, y)}2 + {fy(x, y)}21/2 .

Peak signal to noise image quality metric (PSNR) ASSUMES KNOWN IDEAL IMAGE k(x, y). Let h(x, y) be degraded version of ideal k(x, y), ∗ ∗ ∗ PSNR = −20 log10{ h − k 2 /255} ∗ ∗∗. PSNR = ∞ if h(x, y) = k(x, y), and PSNR decreases monotonically as h(x, y) diverges from k(x, y).

wt = γr(w)∇.{q(x, y, t)∇w}+awwx+b(wcos2w)wy

r(w) = exp(0.025w), q(x, y, t) = e10t(1 + 5e2ysinπx).

NONLINEAR PARABOLIC BLURRING OF SHARP MRI BRAIN IMAGE (A) Original sharp image (B) Blur with a=b=0 (C) Blur with a=1.25, b=0.6

Behavior in above nonlinear blurring Image f 1 ∇f 1 PSNR Sharp A 59 3360 ∞ Blurred B 55 1740 25 Blurred C 55 1770 20

wt = γr(w)∇.{q(x, y, t)∇w} + awwx + b(wcos2w)wy

(B) Blur with a=b=0 (C) Blur with a=1.25, b=0.6 After 10 VanCittert iterns. After 100 VanCittert iterns.

NONLINEAR DEBLURRING OF MRI BRAIN IMAGE

Behavior in above nonlinear deblurring Image f 1 ∇f 1 PSNR Deblurred B 59 2980 34 Deblurred C 63 4780 17

wt = γr(w)∇.{q(x, y, t)∇w}+awwx+b(wcos2w)wy

NONLINEAR PARABOLIC BLURRING OF SHARP FACE IMAGE (D) Original sharp image (E) Blur with a=0, b=0.6 (F) Blur with a=0.83, b=0.6

Behavior in above nonlinear blurring Image f 1 ∇f 1 PSNR Sharp D 107 3100 ∞ Blurred E 101 1580 24 Blurred F 101 1550 20

wt = γr(w)∇.{q(x, y, t)∇w} + awwx + b(wcos2w)wy

(E) Blur with a=0, b=0.6 (F) Blur with a=0.83, b=0.6 After 20 VanCittert iterns. After 100 VanCittert iterns.

NONLINEAR DEBLURRING OF FACE IMAGE

Behavior in above nonlinear deblurring Image f 1 ∇f 1 PSNR Deblurred E 106 2580 29 Deblurred F 112 5800 18

wt = γs(w)∇.{q(x, y, t)∇w}+c

• |w|wx+d(wcos2w)wy

s(w) = 1.0 + 0.00125w2, q = e10t(1 + 5e2ysinπx).

NONLINEAR PARABOLIC BLURRING OF SHARP CARRIER IMAGE (G) Original sharp image (H) Blur with c=2.5, d=0.3 (I) Blur with c=2.5, d=1.5

Behavior in above nonlinear blurring Image f 1 ∇f 1 PSNR Sharp G 139 4760 ∞ Blurred H 134 1720 20 Blurred I 134 1770 20

wt = γs(w)∇.{q(x, y, t)∇w} + c

• |w|wx + d(wcos2w)wy

(H) Blur with c=2.5, d=0.3 (I) Blur with c=2.5, d=1.5 After 100 VanCittert iterns. After 100 VanCittert iterns.

NONLINEAR DEBLURRING OF CARRIER IMAGE

Behavior in above nonlinear deblurring Image f 1 ∇f 1 PSNR Deblurred H 137 3700 23 Deblurred I 141 7700 18

wt = γs(w)∇.{q(x, y, t)∇w} + c

• |w|wx + d(wcos2w)wy

DEBLURRING CARRIER IMAGE WITH FALSE VALUES

After 100 VanCittert iterations Use false c=2.5, d=0.3, to deblur image blurred with c=2.5, d=1.5. FAILED BLIND DECONVOLUTION EXPERIMENT

Original M51 After Linnik deblur Blind deconvolution of KITT PEAK M51 image

LINEAR NONSELFADJOINT PARABOLIC PDE

wt = γr∇.{q(x, y, t)∇w} + awx + bwy

r = 30, q = e10t(1 + 5e2ysinπx) a = 65, b = 35.

LINEAR NONSELFADJOINT PARABOLIC BLURRING Original sharp Sydney image Blurred Sydney image

LINEAR NONSELFADJOINT PARABOLIC PDE

wt = γr∇.{q(x, y, t)∇w} + awx + bwy

r = 30, q = e10t(1 + 5e2ysinπx) a = 65, b = 35.

Blurred Sydney image After 20 VanCittert iterations LINEAR NONSELFADJOINT PARABOLIC DEBLURRING